Problem: Given $ \overrightarrow{OL}\perp\overrightarrow{ON}$, $ m \angle MON = 3x - 12$, and $ m \angle LOM = 8x - 96$, find $m\angle LOM$. $O$ $L$ $N$ $M$
Solution: From the diagram, we see that together ${\angle LOM}$ and ${\angle MON}$ form ${\angle LON}$ , so $ {m\angle LOM} + {m\angle MON} = {m\angle LON}$ Since we are given that $\overrightarrow{OL}\perp\overrightarrow{ON}$ , we know ${m\angle LON = 90}$ Substitute in the expressions that were given for each measure: $ {8x - 96} + {3x - 12} = {90}$ Combine like terms: $ 11x - 108 = 90$ Add $108$ to both sides: $ 11x = 198$ Divide both sides by $11$ to find $x$ $ x = 18$ Substitute $18$ for $x$ in the expression that was given for $m\angle LOM$ $ m\angle LOM = 8({18}) - 96$ Simplify: $ {m\angle LOM = 144 - 96}$ So ${m\angle LOM = 48}$.